Enhanced Geological Prediction for Tunnel Excavation Using Full Waveform Inversion Integrating Sobolev Space Regularization with a Quadratic Penalty Method

In the process of tunnel excavation, advanced geological prediction technology has become indispensable for safe, economical, and efficient tunnel construction. Although traditional methods such as drilling and geological analysis are effective, they typically involve destructive processes, carry high risks, and incur significant costs. In contrast, non-destructive geophysical exploration offers a more convenient and economical alternative. However, the accuracy and precision of these non-destructive methods can be severely compromised by complex geological structures and environmental noise. To address these challenges effectively, a novel approach using frequency domain full waveform inversion (FWI), based on a penalty method and Sobolev space regularization, has been proposed to enhance the performance of non-destructive predictions. The proposed method constructs a soft-constrained optimization problem by restructuring the misfit function into a combination of data misfit and wave equation drive terms to enhance convexity. Additionally, it semi-extends the search space to both the wavefield and the model parameters to mitigate the strong nonlinearity of the optimization, facilitating high-resolution inversion. Furthermore, a Sobolev space regularization algorithm is introduced to flexibly adjust the regularization path, addressing issues related to noise and artefacts to improve the robustness of the inversion. We evaluated the proposed FWI with a tunnel fault model by comparing the results of the proposed method with those of traditional Tikhonov regularization and total variation regularization FWI methods. The results confirm the superior performance of the proposed algorithm as expected.